Warm-Up : EOC Prep

1. Warm-Up:EOC Prep What is RD if RL=54 cm? A) 81 B) 108 C) 162 D) 216 What is x if WL=15x and LJ=5x+3? A) 0.3 B) 0.4 C) 0.6 D) 1.2

2. Inverses, Contrapositives, and Indirect Reasoning5.4 Today’s GoalsBy the end of class today, YOU should be able to… 1. Write the negation of a statement and the inverse and contrapositive of a conditional statement. 2. Use indirect reasoning to solve a problem.

3. Negations • A negation of a statement has the opposite meaning of the original statement.

4. Ex.1: Negations Write the negation of the following statement… Statement: ABC is obtuse.

5. Ex.1: Solution Negation: ABC is not obtuse.

6. You Try… Write the negation of the following statement… Statement: Lines m and n are not perpendicular. 


7. Inverses and contrapositives • A conditional statement is a statement that tells if one thing happens, another will follow • It is written as p q • The inverse of a conditional statement negates both the hypothesis and the conclusion. • The contrapositive of a conditional statement switches both the hypothesis and conclusion and negates both.

8. Ex.2: Write the inverse and contrapositive of the following statement: Statement: If a polygon has six sides, then it is a hexagon.

9. Ex.2: Solution Conditional: If a polygon has six sides, then it is a hexagon.  Negate both.  Inverse: If a polygon does not have six sides, then it is not a hexagon. Conditional: If a polygon has six sides, then it is a hexagon.  Switch and negate both.  Contrapositive: If a polygon is not a hexagon, then it does not have six sides.

10. You Try… Write the inverse and contrapositive of the following statement: If a polygon is a triangle, then the sum of the measures of its angles is 180.

11. A summary of negation, inverse, and contrapositive statements

12. Indirect reasoning • Indirect reasoning is when all possibilities are considered and then all but one are proved false. Then the remaining possibility must be true. • A proof involving indirect reasoning is an indirect proof.

13. An example of indirect reasoning… Suppose your brother tells you, "Susan called a few minutes ago." You think through these three steps. • You have two friends named Susan. • You know that one of them is at band practice. • You conclude that the other Susan must have been the caller.

14. Writing an indirect proof • State as an assumption the opposite (negation) of what you want to prove. • Show that this assumption leads to a contradiction. • Conclude that the assumption must be false and that what you want to prove must be true.

15. Ex.3: Indirect proofs Write the first step of an indirect proof: Quadrilateral QRWX does not have four acute angles.

16. Ex.3: Solution Quadrilateral QRWX does not have four acute angles. Assume that quadrilateral QRWX has four acute angles.

17. You Try… Write the first step of an indirect proof: An integer n is divisible by 5.

18. Ex.4: Indirect proofs Read the conditional statement. Think about what is given and what you are to prove. Then give the steps of an indirect proof. If Jaeleen spends more than $50 to buy two items at a bicycle shop, then at least one of the items costs more than $25.

19. Ex.4: Solution Given: The cost of two items is more than $50. This means that each item costs $25 or less. Prove: At least one of the items costs more than $25. Step 1: Assume as true the opposite of what you want to prove. That is, assume that neither item costs more than $25. Step 2: This, in turn, means that the two items together cost $50 or less. This contradicts the given information that the amount spent is more than $50. Step 3: Conclude that the assumption is false. One item must cost more than $25.

20. Homework • Page 267 #s 8, 13, 15, 16 • Page 268 #s 21, 24-27, 31 • The assignment can also be found online at: • http://www.pearsonsuccessnet.com/snpapp/iText/products/0-13-037878-X/Ch05/05-01/PH_Geom_ch05-01_Ex.pdf